NCERT Textbook: Kinetic Theory | Physics Class 11 - NEET PDF Download

 Page 1


CHAPTER TWELVE
KINETIC THEORY
12.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle,
Newton and several others tried to explain the behaviour of
gases by considering that gases are made up of tiny atomic
particles. The actual atomic theory got established more than
150 years later. Kinetic theory explains the behaviour of gases
based on the idea that the gas  consists of rapidly moving
atoms or molecules. This is possible as the inter-atomic forces,
which are short range forces that are important for solids
and liquids,  can be neglected for gases. The kinetic theory
was developed in the nineteenth century by Maxwell,
Boltzmann and others. It has been remarkably successful. It
gives a molecular interpretation of  pressure and temperature
of a gas, and is consistent with gas laws and Avogadro’s
hypothesis. It correctly explains specific heat capacities of
many gases. It also relates measurable properties of gases
such as viscosity, conduction and diffusion with molecular
parameters, yielding estimates of molecular sizes and masses.
This chapter gives an introduction to kinetic theory.
12.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century
considers the discovery that “Matter is made up of atoms” to
be a very significant one. Humanity may suffer annihilation
(due to nuclear catastrophe) or extinction (due to
environmental disasters) if we do not act wisely. If that
happens, and all of scientific knowledge were to be destroyed
then Feynman would like the ‘Atomic Hypothesis’ to be
communicated to the next generation of creatures in the
universe. Atomic Hypothesis: All things are made of atoms -
little particles that move around in perpetual motion,
attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in
many places and cultures. Kanada in India and Democritus
12.1 Introduction
12.2 Molecular nature of matter
12.3 Behaviour of gases
12.4 Kinetic theory of an ideal gas
12.5 Law of equipartition of energy
12.6 Specific heat capacity
12.7 Mean free path
Summary
Points to ponder
Exercises
2024-25
Page 2


CHAPTER TWELVE
KINETIC THEORY
12.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle,
Newton and several others tried to explain the behaviour of
gases by considering that gases are made up of tiny atomic
particles. The actual atomic theory got established more than
150 years later. Kinetic theory explains the behaviour of gases
based on the idea that the gas  consists of rapidly moving
atoms or molecules. This is possible as the inter-atomic forces,
which are short range forces that are important for solids
and liquids,  can be neglected for gases. The kinetic theory
was developed in the nineteenth century by Maxwell,
Boltzmann and others. It has been remarkably successful. It
gives a molecular interpretation of  pressure and temperature
of a gas, and is consistent with gas laws and Avogadro’s
hypothesis. It correctly explains specific heat capacities of
many gases. It also relates measurable properties of gases
such as viscosity, conduction and diffusion with molecular
parameters, yielding estimates of molecular sizes and masses.
This chapter gives an introduction to kinetic theory.
12.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century
considers the discovery that “Matter is made up of atoms” to
be a very significant one. Humanity may suffer annihilation
(due to nuclear catastrophe) or extinction (due to
environmental disasters) if we do not act wisely. If that
happens, and all of scientific knowledge were to be destroyed
then Feynman would like the ‘Atomic Hypothesis’ to be
communicated to the next generation of creatures in the
universe. Atomic Hypothesis: All things are made of atoms -
little particles that move around in perpetual motion,
attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in
many places and cultures. Kanada in India and Democritus
12.1 Introduction
12.2 Molecular nature of matter
12.3 Behaviour of gases
12.4 Kinetic theory of an ideal gas
12.5 Law of equipartition of energy
12.6 Specific heat capacity
12.7 Mean free path
Summary
Points to ponder
Exercises
2024-25
KINETIC THEORY 245
in Greece had suggested that matter may consist
of indivisible constituents. The scientific ‘Atomic
Theory’  is usually credited to John Dalton. He
proposed the atomic  theory to explain the laws
of definite and multiple proportions obeyed by
elements when they combine into compounds.
The first law says that any given compound has,
a fixed proportion by mass of its constituents.
The second law says that when two elements
form more than one compound, for a fixed mass
of one element, the masses of the other elements
are in ratio of small integers.
To explain the laws Dalton suggested, about
200 years ago,  that the smallest constituents
of an element are atoms. Atoms of one element
are identical but differ from those of other
elements.  A small number of atoms of each
element combine to form a molecule of the
compound. Gay Lussac’s law, also given in early
19
th
 century, states:  When gases combine
chemically to yield another gas, their volumes
are in the ratios of small integers.  Avogadro’s
law  (or hypothesis) says: Equal volumes of all
gases at equal temperature and pressure have
the same number of molecules.  Avogadro’s law,
when combined with Dalton’s theory explains
Gay  Lussac’s law.  Since the elements are often
in the form of molecules, Dalton’s atomic theory
can also be referred to as the molecular theory
of matter. The theory is now well accepted by
scientists. However even at the end of the
nineteenth century there were famous scientists
who did not believe in atomic theory !
From many observations, in recent times we
now know that  molecules (made up of one or
more atoms) constitute matter. Electron
microscopes  and scanning tunnelling
microscopes enable us to even see them. The
size of an atom is about an angstrom (10 
-10
   m).
In solids, which are tightly packed, atoms are
spaced about a few  angstroms (2 Å) apart. In
liquids the separation between atoms is also
about the same.  In liquids the atoms  are not
as rigidly fixed as in solids, and can move
around. This enables a liquid to flow.  In gases
the interatomic distances are in tens of
angstroms.  The average distance a molecule
can travel without colliding is called the  mean
free path. The mean free path, in gases, is of
the order of thousands of angstroms. The atoms
are much freer in gases and can travel long
distances without colliding. If they are not
enclosed, gases disperse away. In solids and
liquids the closeness makes the interatomic force
important. The force has a long range attraction
and a short range repulsion. The atoms attract
when they are at a few angstroms but repel when
they come closer. The static appearance of a gas
Atomic Hypothesis in Ancient India and Greece
Though John Dalton is credited with the introduction of atomic viewpoint in modern science, scholars in
ancient India and Greece conjectured long before the existence of atoms and molecules.  In the Vaiseshika
school of thought in India founded by Kanada (Sixth century B.C.) the atomic picture was developed in
considerable detail. Atoms were thought to be eternal, indivisible, infinitesimal and ultimate parts of matter.
It was argued that if matter could be subdivided without an end, there would be no difference between a
mustard seed and the Meru mountain.  The four kinds of atoms (Paramanu — Sanskrit word for the
smallest particle) postulated were Bhoomi (Earth), Ap (water), Tejas (fire) and Vayu (air) that have characteristic
mass and other attributes, were propounded. Akasa (space) was thought to have no atomic structure and
was continuous and inert. Atoms combine to form different molecules (e.g. two atoms combine to form a
diatomic molecule dvyanuka, three atoms form a tryanuka or a triatomic molecule), their properties depending
upon the nature and ratio of the constituent atoms.  The size of the atoms was also estimated, by conjecture
or by methods that are not known to us.  The estimates vary. In Lalitavistara, a famous biography of the
Buddha written mainly in the second century B.C., the estimate is close to the modern estimate of atomic
size, of the order of 10
–10 
m.
   In ancient Greece, Democritus (Fourth century B.C.) is best known for his atomic hypothesis. The
word ‘atom’ means ‘indivisible’ in Greek. According to him, atoms differ from each other physically, in
shape, size and other properties and this resulted in the different properties of the substances formed
by their combination.  The atoms of water were smooth and round and unable to ‘hook’ on to each
other, which is why liquid /water flows easily.   The atoms of earth were rough and jagged, so they held
together to form hard substances.  The atoms of fire were thorny which is why it caused painful burns.
These fascinating ideas, despite their ingenuity, could not evolve much further, perhaps because they
were intuitive conjectures and speculations not tested and modified by quantitative experiments - the
hallmark of modern science.
2024-25
Page 3


CHAPTER TWELVE
KINETIC THEORY
12.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle,
Newton and several others tried to explain the behaviour of
gases by considering that gases are made up of tiny atomic
particles. The actual atomic theory got established more than
150 years later. Kinetic theory explains the behaviour of gases
based on the idea that the gas  consists of rapidly moving
atoms or molecules. This is possible as the inter-atomic forces,
which are short range forces that are important for solids
and liquids,  can be neglected for gases. The kinetic theory
was developed in the nineteenth century by Maxwell,
Boltzmann and others. It has been remarkably successful. It
gives a molecular interpretation of  pressure and temperature
of a gas, and is consistent with gas laws and Avogadro’s
hypothesis. It correctly explains specific heat capacities of
many gases. It also relates measurable properties of gases
such as viscosity, conduction and diffusion with molecular
parameters, yielding estimates of molecular sizes and masses.
This chapter gives an introduction to kinetic theory.
12.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century
considers the discovery that “Matter is made up of atoms” to
be a very significant one. Humanity may suffer annihilation
(due to nuclear catastrophe) or extinction (due to
environmental disasters) if we do not act wisely. If that
happens, and all of scientific knowledge were to be destroyed
then Feynman would like the ‘Atomic Hypothesis’ to be
communicated to the next generation of creatures in the
universe. Atomic Hypothesis: All things are made of atoms -
little particles that move around in perpetual motion,
attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in
many places and cultures. Kanada in India and Democritus
12.1 Introduction
12.2 Molecular nature of matter
12.3 Behaviour of gases
12.4 Kinetic theory of an ideal gas
12.5 Law of equipartition of energy
12.6 Specific heat capacity
12.7 Mean free path
Summary
Points to ponder
Exercises
2024-25
KINETIC THEORY 245
in Greece had suggested that matter may consist
of indivisible constituents. The scientific ‘Atomic
Theory’  is usually credited to John Dalton. He
proposed the atomic  theory to explain the laws
of definite and multiple proportions obeyed by
elements when they combine into compounds.
The first law says that any given compound has,
a fixed proportion by mass of its constituents.
The second law says that when two elements
form more than one compound, for a fixed mass
of one element, the masses of the other elements
are in ratio of small integers.
To explain the laws Dalton suggested, about
200 years ago,  that the smallest constituents
of an element are atoms. Atoms of one element
are identical but differ from those of other
elements.  A small number of atoms of each
element combine to form a molecule of the
compound. Gay Lussac’s law, also given in early
19
th
 century, states:  When gases combine
chemically to yield another gas, their volumes
are in the ratios of small integers.  Avogadro’s
law  (or hypothesis) says: Equal volumes of all
gases at equal temperature and pressure have
the same number of molecules.  Avogadro’s law,
when combined with Dalton’s theory explains
Gay  Lussac’s law.  Since the elements are often
in the form of molecules, Dalton’s atomic theory
can also be referred to as the molecular theory
of matter. The theory is now well accepted by
scientists. However even at the end of the
nineteenth century there were famous scientists
who did not believe in atomic theory !
From many observations, in recent times we
now know that  molecules (made up of one or
more atoms) constitute matter. Electron
microscopes  and scanning tunnelling
microscopes enable us to even see them. The
size of an atom is about an angstrom (10 
-10
   m).
In solids, which are tightly packed, atoms are
spaced about a few  angstroms (2 Å) apart. In
liquids the separation between atoms is also
about the same.  In liquids the atoms  are not
as rigidly fixed as in solids, and can move
around. This enables a liquid to flow.  In gases
the interatomic distances are in tens of
angstroms.  The average distance a molecule
can travel without colliding is called the  mean
free path. The mean free path, in gases, is of
the order of thousands of angstroms. The atoms
are much freer in gases and can travel long
distances without colliding. If they are not
enclosed, gases disperse away. In solids and
liquids the closeness makes the interatomic force
important. The force has a long range attraction
and a short range repulsion. The atoms attract
when they are at a few angstroms but repel when
they come closer. The static appearance of a gas
Atomic Hypothesis in Ancient India and Greece
Though John Dalton is credited with the introduction of atomic viewpoint in modern science, scholars in
ancient India and Greece conjectured long before the existence of atoms and molecules.  In the Vaiseshika
school of thought in India founded by Kanada (Sixth century B.C.) the atomic picture was developed in
considerable detail. Atoms were thought to be eternal, indivisible, infinitesimal and ultimate parts of matter.
It was argued that if matter could be subdivided without an end, there would be no difference between a
mustard seed and the Meru mountain.  The four kinds of atoms (Paramanu — Sanskrit word for the
smallest particle) postulated were Bhoomi (Earth), Ap (water), Tejas (fire) and Vayu (air) that have characteristic
mass and other attributes, were propounded. Akasa (space) was thought to have no atomic structure and
was continuous and inert. Atoms combine to form different molecules (e.g. two atoms combine to form a
diatomic molecule dvyanuka, three atoms form a tryanuka or a triatomic molecule), their properties depending
upon the nature and ratio of the constituent atoms.  The size of the atoms was also estimated, by conjecture
or by methods that are not known to us.  The estimates vary. In Lalitavistara, a famous biography of the
Buddha written mainly in the second century B.C., the estimate is close to the modern estimate of atomic
size, of the order of 10
–10 
m.
   In ancient Greece, Democritus (Fourth century B.C.) is best known for his atomic hypothesis. The
word ‘atom’ means ‘indivisible’ in Greek. According to him, atoms differ from each other physically, in
shape, size and other properties and this resulted in the different properties of the substances formed
by their combination.  The atoms of water were smooth and round and unable to ‘hook’ on to each
other, which is why liquid /water flows easily.   The atoms of earth were rough and jagged, so they held
together to form hard substances.  The atoms of fire were thorny which is why it caused painful burns.
These fascinating ideas, despite their ingenuity, could not evolve much further, perhaps because they
were intuitive conjectures and speculations not tested and modified by quantitative experiments - the
hallmark of modern science.
2024-25
246 PHYSICS
( )
–1 –1
Jmol K
pV
T µ
is misleading. The gas is full of activity and the
equilibrium is a dynamic one. In dynamic
equilibrium, molecules collide and change their
speeds during the collision. Only the average
properties are constant.
Atomic theory is not the end of our quest, but
the beginning. We now know that atoms are not
indivisible or elementary. They consist of a
nucleus and electrons. The nucleus itself is made
up of protons and neutrons. The protons and
neutrons are again made up of quarks. Even
quarks may not be the end of the story. There
may be string like elementary entities. Nature
always has surprises for us, but the search for
truth is often enjoyable and the discoveries
beautiful. In this chapter, we shall limit ourselves
to understanding the behaviour of gases (and a
little bit of solids), as a collection of moving
molecules in incessant motion.
12.3   BEHAVIOUR OF GASES
Properties of gases are easier to understand than
those of solids and liquids. This is mainly
because in a gas, molecules are far from each
other and their mutual interactions are
negligible except when two molecules collide.
Gases at low pressures and high temperatures
much above that at which they liquefy (or
solidify) approximately satisfy a simple relation
between their pressure, temperature and volume
given by (see Chapter 10)
PV = KT (12.1)
for a given sample of the gas. Here T is the
temperature in kelvin or (absolute)  scale. K is  a
constant for the given sample but varies with
the volume of the gas. If we now  bring in  the
idea of atoms or molecules, then K is proportional
to the number of molecules, (say) N in the
sample. We can write K = N k . Observation tells
us that this k is same for all gases. It is called
Boltzmann constant and is denoted by k
B
.
As 
1 1 2 2
1 1 2 2
PV P V
N T N T
= = constant = k
B
(12.2)
if P, V and T are same, then N is also same for all
gases. This is Avogadro’s hypothesis, that  the
number of molecules per unit volume is
the same for all gases at a fixed temperature and
pressure. The number in 22.4 litres of any gas
is 6.02 × 10
23
. This is known as Avogadro number
and is denoted by N
A
. The mass of 22.4 litres of
any gas is equal to its molecular weight in grams
at S.T.P (standard temperature 273 K and
pressure 1 atm). This amount of substance is
called a mole (see Chapter 1 for a more precise
definition). Avogadro had guessed the equality of
numbers in equal volumes of gas at a fixed
temperature and pressure from chemical
reactions.  Kinetic  theory justifies this hypothesis.
The perfect gas equation can be written as
PV = µ RT (12.3)
where  µ   is the number of moles and R  = N
A
k
B
 is a universal constant. The temperature T is
absolute temperature.  Choosing kelvin scale for
absolute temperature, R = 8.314 J mol
–1
K
–1
.
Here
0 A
M N
M N
µ = =
(12.4)
where M is the mass of the gas containing N
molecules, M
0
 is the molar mass and N
A
 the
Avogadro’s number. Using  Eqs. (12.4) and (12.3)
can also be written as
PV = k
B
 NT or P = k
B
 nT
P (atm)
Fig.12.1 Real gases approach ideal gas behaviour at
low pressures and high temperatures.
where  n is the number density, i.e. number of
molecules per unit volume. k
B
 is  the Boltzmann
constant introduced above. Its value in SI units
is 1.38 × 10
–23
 J K
–1
.
Another useful form of Eq. (12.3) is
0
RT
P
M
?
=
(12.5)
2024-25
Page 4


CHAPTER TWELVE
KINETIC THEORY
12.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle,
Newton and several others tried to explain the behaviour of
gases by considering that gases are made up of tiny atomic
particles. The actual atomic theory got established more than
150 years later. Kinetic theory explains the behaviour of gases
based on the idea that the gas  consists of rapidly moving
atoms or molecules. This is possible as the inter-atomic forces,
which are short range forces that are important for solids
and liquids,  can be neglected for gases. The kinetic theory
was developed in the nineteenth century by Maxwell,
Boltzmann and others. It has been remarkably successful. It
gives a molecular interpretation of  pressure and temperature
of a gas, and is consistent with gas laws and Avogadro’s
hypothesis. It correctly explains specific heat capacities of
many gases. It also relates measurable properties of gases
such as viscosity, conduction and diffusion with molecular
parameters, yielding estimates of molecular sizes and masses.
This chapter gives an introduction to kinetic theory.
12.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century
considers the discovery that “Matter is made up of atoms” to
be a very significant one. Humanity may suffer annihilation
(due to nuclear catastrophe) or extinction (due to
environmental disasters) if we do not act wisely. If that
happens, and all of scientific knowledge were to be destroyed
then Feynman would like the ‘Atomic Hypothesis’ to be
communicated to the next generation of creatures in the
universe. Atomic Hypothesis: All things are made of atoms -
little particles that move around in perpetual motion,
attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in
many places and cultures. Kanada in India and Democritus
12.1 Introduction
12.2 Molecular nature of matter
12.3 Behaviour of gases
12.4 Kinetic theory of an ideal gas
12.5 Law of equipartition of energy
12.6 Specific heat capacity
12.7 Mean free path
Summary
Points to ponder
Exercises
2024-25
KINETIC THEORY 245
in Greece had suggested that matter may consist
of indivisible constituents. The scientific ‘Atomic
Theory’  is usually credited to John Dalton. He
proposed the atomic  theory to explain the laws
of definite and multiple proportions obeyed by
elements when they combine into compounds.
The first law says that any given compound has,
a fixed proportion by mass of its constituents.
The second law says that when two elements
form more than one compound, for a fixed mass
of one element, the masses of the other elements
are in ratio of small integers.
To explain the laws Dalton suggested, about
200 years ago,  that the smallest constituents
of an element are atoms. Atoms of one element
are identical but differ from those of other
elements.  A small number of atoms of each
element combine to form a molecule of the
compound. Gay Lussac’s law, also given in early
19
th
 century, states:  When gases combine
chemically to yield another gas, their volumes
are in the ratios of small integers.  Avogadro’s
law  (or hypothesis) says: Equal volumes of all
gases at equal temperature and pressure have
the same number of molecules.  Avogadro’s law,
when combined with Dalton’s theory explains
Gay  Lussac’s law.  Since the elements are often
in the form of molecules, Dalton’s atomic theory
can also be referred to as the molecular theory
of matter. The theory is now well accepted by
scientists. However even at the end of the
nineteenth century there were famous scientists
who did not believe in atomic theory !
From many observations, in recent times we
now know that  molecules (made up of one or
more atoms) constitute matter. Electron
microscopes  and scanning tunnelling
microscopes enable us to even see them. The
size of an atom is about an angstrom (10 
-10
   m).
In solids, which are tightly packed, atoms are
spaced about a few  angstroms (2 Å) apart. In
liquids the separation between atoms is also
about the same.  In liquids the atoms  are not
as rigidly fixed as in solids, and can move
around. This enables a liquid to flow.  In gases
the interatomic distances are in tens of
angstroms.  The average distance a molecule
can travel without colliding is called the  mean
free path. The mean free path, in gases, is of
the order of thousands of angstroms. The atoms
are much freer in gases and can travel long
distances without colliding. If they are not
enclosed, gases disperse away. In solids and
liquids the closeness makes the interatomic force
important. The force has a long range attraction
and a short range repulsion. The atoms attract
when they are at a few angstroms but repel when
they come closer. The static appearance of a gas
Atomic Hypothesis in Ancient India and Greece
Though John Dalton is credited with the introduction of atomic viewpoint in modern science, scholars in
ancient India and Greece conjectured long before the existence of atoms and molecules.  In the Vaiseshika
school of thought in India founded by Kanada (Sixth century B.C.) the atomic picture was developed in
considerable detail. Atoms were thought to be eternal, indivisible, infinitesimal and ultimate parts of matter.
It was argued that if matter could be subdivided without an end, there would be no difference between a
mustard seed and the Meru mountain.  The four kinds of atoms (Paramanu — Sanskrit word for the
smallest particle) postulated were Bhoomi (Earth), Ap (water), Tejas (fire) and Vayu (air) that have characteristic
mass and other attributes, were propounded. Akasa (space) was thought to have no atomic structure and
was continuous and inert. Atoms combine to form different molecules (e.g. two atoms combine to form a
diatomic molecule dvyanuka, three atoms form a tryanuka or a triatomic molecule), their properties depending
upon the nature and ratio of the constituent atoms.  The size of the atoms was also estimated, by conjecture
or by methods that are not known to us.  The estimates vary. In Lalitavistara, a famous biography of the
Buddha written mainly in the second century B.C., the estimate is close to the modern estimate of atomic
size, of the order of 10
–10 
m.
   In ancient Greece, Democritus (Fourth century B.C.) is best known for his atomic hypothesis. The
word ‘atom’ means ‘indivisible’ in Greek. According to him, atoms differ from each other physically, in
shape, size and other properties and this resulted in the different properties of the substances formed
by their combination.  The atoms of water were smooth and round and unable to ‘hook’ on to each
other, which is why liquid /water flows easily.   The atoms of earth were rough and jagged, so they held
together to form hard substances.  The atoms of fire were thorny which is why it caused painful burns.
These fascinating ideas, despite their ingenuity, could not evolve much further, perhaps because they
were intuitive conjectures and speculations not tested and modified by quantitative experiments - the
hallmark of modern science.
2024-25
246 PHYSICS
( )
–1 –1
Jmol K
pV
T µ
is misleading. The gas is full of activity and the
equilibrium is a dynamic one. In dynamic
equilibrium, molecules collide and change their
speeds during the collision. Only the average
properties are constant.
Atomic theory is not the end of our quest, but
the beginning. We now know that atoms are not
indivisible or elementary. They consist of a
nucleus and electrons. The nucleus itself is made
up of protons and neutrons. The protons and
neutrons are again made up of quarks. Even
quarks may not be the end of the story. There
may be string like elementary entities. Nature
always has surprises for us, but the search for
truth is often enjoyable and the discoveries
beautiful. In this chapter, we shall limit ourselves
to understanding the behaviour of gases (and a
little bit of solids), as a collection of moving
molecules in incessant motion.
12.3   BEHAVIOUR OF GASES
Properties of gases are easier to understand than
those of solids and liquids. This is mainly
because in a gas, molecules are far from each
other and their mutual interactions are
negligible except when two molecules collide.
Gases at low pressures and high temperatures
much above that at which they liquefy (or
solidify) approximately satisfy a simple relation
between their pressure, temperature and volume
given by (see Chapter 10)
PV = KT (12.1)
for a given sample of the gas. Here T is the
temperature in kelvin or (absolute)  scale. K is  a
constant for the given sample but varies with
the volume of the gas. If we now  bring in  the
idea of atoms or molecules, then K is proportional
to the number of molecules, (say) N in the
sample. We can write K = N k . Observation tells
us that this k is same for all gases. It is called
Boltzmann constant and is denoted by k
B
.
As 
1 1 2 2
1 1 2 2
PV P V
N T N T
= = constant = k
B
(12.2)
if P, V and T are same, then N is also same for all
gases. This is Avogadro’s hypothesis, that  the
number of molecules per unit volume is
the same for all gases at a fixed temperature and
pressure. The number in 22.4 litres of any gas
is 6.02 × 10
23
. This is known as Avogadro number
and is denoted by N
A
. The mass of 22.4 litres of
any gas is equal to its molecular weight in grams
at S.T.P (standard temperature 273 K and
pressure 1 atm). This amount of substance is
called a mole (see Chapter 1 for a more precise
definition). Avogadro had guessed the equality of
numbers in equal volumes of gas at a fixed
temperature and pressure from chemical
reactions.  Kinetic  theory justifies this hypothesis.
The perfect gas equation can be written as
PV = µ RT (12.3)
where  µ   is the number of moles and R  = N
A
k
B
 is a universal constant. The temperature T is
absolute temperature.  Choosing kelvin scale for
absolute temperature, R = 8.314 J mol
–1
K
–1
.
Here
0 A
M N
M N
µ = =
(12.4)
where M is the mass of the gas containing N
molecules, M
0
 is the molar mass and N
A
 the
Avogadro’s number. Using  Eqs. (12.4) and (12.3)
can also be written as
PV = k
B
 NT or P = k
B
 nT
P (atm)
Fig.12.1 Real gases approach ideal gas behaviour at
low pressures and high temperatures.
where  n is the number density, i.e. number of
molecules per unit volume. k
B
 is  the Boltzmann
constant introduced above. Its value in SI units
is 1.38 × 10
–23
 J K
–1
.
Another useful form of Eq. (12.3) is
0
RT
P
M
?
=
(12.5)
2024-25
KINETIC THEORY 247
?
where ? is the mass density of the gas.
A gas that satisfies Eq. (12.3) exactly at all
pressures and temperatures is defined to be an
ideal gas. An ideal gas is a simple theoretical
model of a gas. No real gas is truly ideal.
Fig. 12.1 shows departures from ideal gas
behaviour for a real gas at three different
temperatures. Notice that all curves approach
the ideal gas behaviour for low  pressures and
high temperatures.
At low pressures or high temperatures the
molecules are far apart and molecular
interactions are negligible. Without interactions
the gas behaves like an ideal one.
If we fix µ and T in Eq. (12.3), we get
PV = constant (12.6)
i.e., keeping temperature constant, pressure of
a given mass of gas varies inversely with volume.
This is the famous Boyle’s law. Fig. 12.2  shows
comparison between experimental P-V curves
and the theoretical curves predicted by Boyle’s
law. Once again you see that the  agreement is
good at high temperatures and  low pressures.
Next, if you fix P, Eq. (12.1) shows that V ?  T i.e.,
for a fixed pressure, the volume of a gas is
proportional to its absolute temperature T
(Charles’ law). See Fig. 12.3.
Fig.12.2 Experimental P-V curves (solid lines) for
steam at three temperatures compared with
Boyle’s law (dotted lines). P is in units of 22
atm and V in units of 0.09 litres.
Finally, consider a mixture of non-interacting
ideal  gases: µ
1
  moles of gas 1, µ
2
 moles of gas 2,
etc. in a vessel of volume V at temperature T and
pressure P. It is then found that the equation  of
state of the mixture is :
PV = ( µ
1
 + µ
2 
+…  ) RT (12.7)
i.e. 
1 2
...
RT RT
P
V V
µ µ = + + (12.8)
= P
1
 + P
2
 + … (12.9)
Clearly P
1
 =    µ
1
 R T/V   is the pressure that
gas 1 would  exert at the same conditions of
volume and  temperature if no other gases were
present. This is called the partial pressure of the
gas. Thus, the total pressure of a mixture of ideal
gases is the sum of partial pressures. This is
Dalton’s law of partial pressures.
Fig. 12.3 Experimental T-V curves (solid lines) for CO
2
at three pressures compared with Charles’
law (dotted lines). T is in units of 300 K and
V in units of 0.13 litres.
We next consider some examples which give
us information about the volume occupied by
the molecules and the volume of a single
molecule.
Example 12.1 The density of water is  1000
kg m
–3
. The density of water vapour at 100 °C
and 1 atm pressure is 0.6 kg m
–3
. The
volume of a molecule multiplied by the total
number gives ,what is called, molecular
volume. Estimate the ratio (or fraction) of
the molecular volume  to the total volume
occupied by the water vapour under the
above conditions of temperature and
pressure.
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Page 5


CHAPTER TWELVE
KINETIC THEORY
12.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle,
Newton and several others tried to explain the behaviour of
gases by considering that gases are made up of tiny atomic
particles. The actual atomic theory got established more than
150 years later. Kinetic theory explains the behaviour of gases
based on the idea that the gas  consists of rapidly moving
atoms or molecules. This is possible as the inter-atomic forces,
which are short range forces that are important for solids
and liquids,  can be neglected for gases. The kinetic theory
was developed in the nineteenth century by Maxwell,
Boltzmann and others. It has been remarkably successful. It
gives a molecular interpretation of  pressure and temperature
of a gas, and is consistent with gas laws and Avogadro’s
hypothesis. It correctly explains specific heat capacities of
many gases. It also relates measurable properties of gases
such as viscosity, conduction and diffusion with molecular
parameters, yielding estimates of molecular sizes and masses.
This chapter gives an introduction to kinetic theory.
12.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century
considers the discovery that “Matter is made up of atoms” to
be a very significant one. Humanity may suffer annihilation
(due to nuclear catastrophe) or extinction (due to
environmental disasters) if we do not act wisely. If that
happens, and all of scientific knowledge were to be destroyed
then Feynman would like the ‘Atomic Hypothesis’ to be
communicated to the next generation of creatures in the
universe. Atomic Hypothesis: All things are made of atoms -
little particles that move around in perpetual motion,
attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in
many places and cultures. Kanada in India and Democritus
12.1 Introduction
12.2 Molecular nature of matter
12.3 Behaviour of gases
12.4 Kinetic theory of an ideal gas
12.5 Law of equipartition of energy
12.6 Specific heat capacity
12.7 Mean free path
Summary
Points to ponder
Exercises
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KINETIC THEORY 245
in Greece had suggested that matter may consist
of indivisible constituents. The scientific ‘Atomic
Theory’  is usually credited to John Dalton. He
proposed the atomic  theory to explain the laws
of definite and multiple proportions obeyed by
elements when they combine into compounds.
The first law says that any given compound has,
a fixed proportion by mass of its constituents.
The second law says that when two elements
form more than one compound, for a fixed mass
of one element, the masses of the other elements
are in ratio of small integers.
To explain the laws Dalton suggested, about
200 years ago,  that the smallest constituents
of an element are atoms. Atoms of one element
are identical but differ from those of other
elements.  A small number of atoms of each
element combine to form a molecule of the
compound. Gay Lussac’s law, also given in early
19
th
 century, states:  When gases combine
chemically to yield another gas, their volumes
are in the ratios of small integers.  Avogadro’s
law  (or hypothesis) says: Equal volumes of all
gases at equal temperature and pressure have
the same number of molecules.  Avogadro’s law,
when combined with Dalton’s theory explains
Gay  Lussac’s law.  Since the elements are often
in the form of molecules, Dalton’s atomic theory
can also be referred to as the molecular theory
of matter. The theory is now well accepted by
scientists. However even at the end of the
nineteenth century there were famous scientists
who did not believe in atomic theory !
From many observations, in recent times we
now know that  molecules (made up of one or
more atoms) constitute matter. Electron
microscopes  and scanning tunnelling
microscopes enable us to even see them. The
size of an atom is about an angstrom (10 
-10
   m).
In solids, which are tightly packed, atoms are
spaced about a few  angstroms (2 Å) apart. In
liquids the separation between atoms is also
about the same.  In liquids the atoms  are not
as rigidly fixed as in solids, and can move
around. This enables a liquid to flow.  In gases
the interatomic distances are in tens of
angstroms.  The average distance a molecule
can travel without colliding is called the  mean
free path. The mean free path, in gases, is of
the order of thousands of angstroms. The atoms
are much freer in gases and can travel long
distances without colliding. If they are not
enclosed, gases disperse away. In solids and
liquids the closeness makes the interatomic force
important. The force has a long range attraction
and a short range repulsion. The atoms attract
when they are at a few angstroms but repel when
they come closer. The static appearance of a gas
Atomic Hypothesis in Ancient India and Greece
Though John Dalton is credited with the introduction of atomic viewpoint in modern science, scholars in
ancient India and Greece conjectured long before the existence of atoms and molecules.  In the Vaiseshika
school of thought in India founded by Kanada (Sixth century B.C.) the atomic picture was developed in
considerable detail. Atoms were thought to be eternal, indivisible, infinitesimal and ultimate parts of matter.
It was argued that if matter could be subdivided without an end, there would be no difference between a
mustard seed and the Meru mountain.  The four kinds of atoms (Paramanu — Sanskrit word for the
smallest particle) postulated were Bhoomi (Earth), Ap (water), Tejas (fire) and Vayu (air) that have characteristic
mass and other attributes, were propounded. Akasa (space) was thought to have no atomic structure and
was continuous and inert. Atoms combine to form different molecules (e.g. two atoms combine to form a
diatomic molecule dvyanuka, three atoms form a tryanuka or a triatomic molecule), their properties depending
upon the nature and ratio of the constituent atoms.  The size of the atoms was also estimated, by conjecture
or by methods that are not known to us.  The estimates vary. In Lalitavistara, a famous biography of the
Buddha written mainly in the second century B.C., the estimate is close to the modern estimate of atomic
size, of the order of 10
–10 
m.
   In ancient Greece, Democritus (Fourth century B.C.) is best known for his atomic hypothesis. The
word ‘atom’ means ‘indivisible’ in Greek. According to him, atoms differ from each other physically, in
shape, size and other properties and this resulted in the different properties of the substances formed
by their combination.  The atoms of water were smooth and round and unable to ‘hook’ on to each
other, which is why liquid /water flows easily.   The atoms of earth were rough and jagged, so they held
together to form hard substances.  The atoms of fire were thorny which is why it caused painful burns.
These fascinating ideas, despite their ingenuity, could not evolve much further, perhaps because they
were intuitive conjectures and speculations not tested and modified by quantitative experiments - the
hallmark of modern science.
2024-25
246 PHYSICS
( )
–1 –1
Jmol K
pV
T µ
is misleading. The gas is full of activity and the
equilibrium is a dynamic one. In dynamic
equilibrium, molecules collide and change their
speeds during the collision. Only the average
properties are constant.
Atomic theory is not the end of our quest, but
the beginning. We now know that atoms are not
indivisible or elementary. They consist of a
nucleus and electrons. The nucleus itself is made
up of protons and neutrons. The protons and
neutrons are again made up of quarks. Even
quarks may not be the end of the story. There
may be string like elementary entities. Nature
always has surprises for us, but the search for
truth is often enjoyable and the discoveries
beautiful. In this chapter, we shall limit ourselves
to understanding the behaviour of gases (and a
little bit of solids), as a collection of moving
molecules in incessant motion.
12.3   BEHAVIOUR OF GASES
Properties of gases are easier to understand than
those of solids and liquids. This is mainly
because in a gas, molecules are far from each
other and their mutual interactions are
negligible except when two molecules collide.
Gases at low pressures and high temperatures
much above that at which they liquefy (or
solidify) approximately satisfy a simple relation
between their pressure, temperature and volume
given by (see Chapter 10)
PV = KT (12.1)
for a given sample of the gas. Here T is the
temperature in kelvin or (absolute)  scale. K is  a
constant for the given sample but varies with
the volume of the gas. If we now  bring in  the
idea of atoms or molecules, then K is proportional
to the number of molecules, (say) N in the
sample. We can write K = N k . Observation tells
us that this k is same for all gases. It is called
Boltzmann constant and is denoted by k
B
.
As 
1 1 2 2
1 1 2 2
PV P V
N T N T
= = constant = k
B
(12.2)
if P, V and T are same, then N is also same for all
gases. This is Avogadro’s hypothesis, that  the
number of molecules per unit volume is
the same for all gases at a fixed temperature and
pressure. The number in 22.4 litres of any gas
is 6.02 × 10
23
. This is known as Avogadro number
and is denoted by N
A
. The mass of 22.4 litres of
any gas is equal to its molecular weight in grams
at S.T.P (standard temperature 273 K and
pressure 1 atm). This amount of substance is
called a mole (see Chapter 1 for a more precise
definition). Avogadro had guessed the equality of
numbers in equal volumes of gas at a fixed
temperature and pressure from chemical
reactions.  Kinetic  theory justifies this hypothesis.
The perfect gas equation can be written as
PV = µ RT (12.3)
where  µ   is the number of moles and R  = N
A
k
B
 is a universal constant. The temperature T is
absolute temperature.  Choosing kelvin scale for
absolute temperature, R = 8.314 J mol
–1
K
–1
.
Here
0 A
M N
M N
µ = =
(12.4)
where M is the mass of the gas containing N
molecules, M
0
 is the molar mass and N
A
 the
Avogadro’s number. Using  Eqs. (12.4) and (12.3)
can also be written as
PV = k
B
 NT or P = k
B
 nT
P (atm)
Fig.12.1 Real gases approach ideal gas behaviour at
low pressures and high temperatures.
where  n is the number density, i.e. number of
molecules per unit volume. k
B
 is  the Boltzmann
constant introduced above. Its value in SI units
is 1.38 × 10
–23
 J K
–1
.
Another useful form of Eq. (12.3) is
0
RT
P
M
?
=
(12.5)
2024-25
KINETIC THEORY 247
?
where ? is the mass density of the gas.
A gas that satisfies Eq. (12.3) exactly at all
pressures and temperatures is defined to be an
ideal gas. An ideal gas is a simple theoretical
model of a gas. No real gas is truly ideal.
Fig. 12.1 shows departures from ideal gas
behaviour for a real gas at three different
temperatures. Notice that all curves approach
the ideal gas behaviour for low  pressures and
high temperatures.
At low pressures or high temperatures the
molecules are far apart and molecular
interactions are negligible. Without interactions
the gas behaves like an ideal one.
If we fix µ and T in Eq. (12.3), we get
PV = constant (12.6)
i.e., keeping temperature constant, pressure of
a given mass of gas varies inversely with volume.
This is the famous Boyle’s law. Fig. 12.2  shows
comparison between experimental P-V curves
and the theoretical curves predicted by Boyle’s
law. Once again you see that the  agreement is
good at high temperatures and  low pressures.
Next, if you fix P, Eq. (12.1) shows that V ?  T i.e.,
for a fixed pressure, the volume of a gas is
proportional to its absolute temperature T
(Charles’ law). See Fig. 12.3.
Fig.12.2 Experimental P-V curves (solid lines) for
steam at three temperatures compared with
Boyle’s law (dotted lines). P is in units of 22
atm and V in units of 0.09 litres.
Finally, consider a mixture of non-interacting
ideal  gases: µ
1
  moles of gas 1, µ
2
 moles of gas 2,
etc. in a vessel of volume V at temperature T and
pressure P. It is then found that the equation  of
state of the mixture is :
PV = ( µ
1
 + µ
2 
+…  ) RT (12.7)
i.e. 
1 2
...
RT RT
P
V V
µ µ = + + (12.8)
= P
1
 + P
2
 + … (12.9)
Clearly P
1
 =    µ
1
 R T/V   is the pressure that
gas 1 would  exert at the same conditions of
volume and  temperature if no other gases were
present. This is called the partial pressure of the
gas. Thus, the total pressure of a mixture of ideal
gases is the sum of partial pressures. This is
Dalton’s law of partial pressures.
Fig. 12.3 Experimental T-V curves (solid lines) for CO
2
at three pressures compared with Charles’
law (dotted lines). T is in units of 300 K and
V in units of 0.13 litres.
We next consider some examples which give
us information about the volume occupied by
the molecules and the volume of a single
molecule.
Example 12.1 The density of water is  1000
kg m
–3
. The density of water vapour at 100 °C
and 1 atm pressure is 0.6 kg m
–3
. The
volume of a molecule multiplied by the total
number gives ,what is called, molecular
volume. Estimate the ratio (or fraction) of
the molecular volume  to the total volume
occupied by the water vapour under the
above conditions of temperature and
pressure.
2024-25
248 PHYSICS
?
?
Answer For  a given mass of water molecules,
the density is less if volume is large. So the
volume of the vapour is  1000/0.6  = 1/(6 × 10 
-4 
)
times larger.  If densities of bulk water and water
molecules are same, then the fraction of
molecular volume to the total volume in liquid
state is 1. As volume in vapour state has
increased, the fractional volume is less by the
same amount, i.e.  6×10
-4
. ?
Example 12.2   Estimate the volume of a
water molecule using the data in Example
12.1.
Answer In the liquid (or solid) phase, the
molecules of water are quite closely packed. The
density of water molecule may therefore, be
regarded as roughly equal to the density of bulk
water = 1000 kg m
–3
. To estimate the volume of
a water molecule, we need to know the mass of
a single water molecule. We know that 1 mole
of water has a mass approximately equal to
(2 + 16)g  = 18 g  =  0.018 kg.
Since 1 mole   contains  about   6 × 10
23
molecules   (Avogadro’s  number),   the mass of
a molecule of water is  (0.018)/(6 × 10
23
) kg  =
3 × 10
–26
 kg. Therefore, a rough estimate of the
volume of a water  molecule is as follows :
Volume of a water molecule
= (3 × 10
–26
 kg)/ (1000 kg m
–3
)
= 3 × 10
–29
 m
3
= (4/3) p  (Radius)
3
Hence, Radius ˜ 2 ×10
-10
  m = 2 Å ?
Example 12.3   What is the average
distance between atoms (interatomic
distance) in water? Use the data given in
Examples 12.1 and 12.2.
Answer :   A given mass of water in vapour state
has 1.67×10
3
 times the volume of the same mass
of water in liquid state (Ex. 12.1). This is also
the increase in the amount of volume available
for each molecule of water. When volume
increases by 10
3
 times the radius increases by
V
1/3
 or 10 times, i.e., 10 × 2 Å  = 20 Å. So the
average distance is 2 × 20 = 40  Å.    ?
Example 12.4 A vessel contains two non-
reactive gases : neon (monatomic) and
oxygen (diatomic). The ratio of their partial
pressures is 3:2. Estimate the ratio of  (i)
number of molecules and (ii) mass density
of neon and oxygen in the vessel. Atomic
mass of Ne = 20.2 u, molecular mass of O
2
= 32.0 u.
Answer Partial pressure of a gas in a mixture is
the pressure it would have for the same volume
and temperature if it alone occupied the vessel.
(The total pressure of a mixture of non-reactive
gases is the sum of partial pressures due to its
constituent gases.) Each gas (assumed ideal)
obeys the gas law. Since V and T are common to
the two gases,  we  have  P
1
V = µ 
1
 RT and P
2
V =
µ
2
 RT, i.e. (P
1
/P
2
) = (µ
1 
/ µ
2
). Here 1 and 2 refer
to neon and oxygen respectively. Since (P
1
/P
2
) =
(3/2) (given), (µ
1
/ µ
2
) = 3/2.
(i) By definition µ
1
 = (N
1
/N
A
 ) and µ
2 
= (N
2
/N
A
)
where N
1
 and N
2
 are the number of molecules
of 1 and 2, and N
A
 is the Avogadro’s number.
Therefore, (N
1
/N
2
) = (µ
1 
/ µ
2
)  = 3/2.
(ii) We can also write µ
1
 = (m
1
/M
1
) and µ
2
 =
(m
2
/M
2
) where m
1 
and m
2
 are the masses of
1 and 2; and M
1
 and M
2
 are their molecular
masses. (Both m
1
 and M
1
; as well as m
2
 and
M
2
 should be expressed in the same units).
If ?
1
 and ?
2 
 are the mass densities of 1 and
2 respectively,  we have
?
?
µ
µ
1
2
1
2
1
2
1
2
1
2
= = = ×
?
?
?
?
?
?
m V
m V
m
m
M
M
/
/
3 20.2
0.947
2 32.0
= × =
  ?
12.4 KINETIC THEORY OF AN IDEAL GAS
Kinetic theory of gases is based on the molecular
picture of matter. A given amount of gas is a
collection of a large number of molecules
(typically of the order of Avogadro’s number) that
are in incessant random motion. At ordinary
pressure and temperature, the average distance
between molecules is a factor of 10 or more than
the typical size of a molecule (2 Å). Thus,
interaction between molecules is negligible and
we can assume that they move freely in straight
lines according to Newton’s first law. However,
occasionally, they come close to each other,
experience intermolecular forces and their
velocities change.  These interactions are called
collisions. The molecules collide incessantly
against each other or with the walls and change
?
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